Question

Find all the real solutions of the equation. \(x^3+4 x^2-11 x-30=0\)

Short answer

The roots of the equation \(x^3 + 4x^2 - 11x - 30 = 0\) are -5, 2 and 3.

Step-by-step solution

Find possible rational roots

We could first apply the rational root theorem. The rational root theorem states that any rational root, written in simplest form \(p/q\), is such that \(p\) is a factor of the constant term (-30) and \(q\) is a factor of the leading coefficient (1). So, the possible rational roots are ±1, ± 2, ± 3, ± 5, ± 6, ± 10, ± 15, ± 30.

Identifying actual roots

Then we would substitute these values into the equation to identify the actual roots. After doing so, we find that -5, 2 and 3 are roots of the equation.

Verification

To confirm that -5, 2 and 3 are indeed the roots of the equation, substitute them back into the equation to get zero.

Key concepts

Polynomial Equations

A polynomial equation is an expression consisting of variables, coefficients, and non-negative integer exponents which is set equal to zero. The equation provides a relationship where the left-hand side must be equivalent to the right-hand side when the variables (often denoted by x) take on certain values. These specific values of x that satisfy the equation are known as solutions or roots of the polynomial.

For instance, in the equation \(x^3+4x^2-11x-30=0\), we have a third-degree polynomial (also called a cubic polynomial) since the highest exponent of x is three. The goal in solving such an equation is to find all values of x that make the equation true. These solutions may be real or complex numbers.

Finding Real Solutions

To find real solutions of polynomial equations, we often rely on various algebraic methods and properties. The Rational Root Theorem is particularly useful for pinpointing potential real rational solutions. It specifies potential rational solutions as factors of the constant term (at the end of the polynomial) divided by factors of the leading coefficient (the number before the term with the highest exponent).

After listing the potential rational solutions, we then substitute these values into the original polynomial equation to check if they indeed satisfy the equation. Not every potential solution will work, so this process of substitution helps us identify the actual rational roots. Once a root is found, it can be used to simplify the equation further and find other solutions, until all real solutions are determined.

Factor Theorem

The Factor Theorem is a special case of the Polynomial Remainder Theorem and provides a quick way to check if a number is a root of a polynomial. According to the Factor Theorem: if substituting a value \(a\) into the polynomial gives a result of zero, \(a\) is a root, and \(x-a\) is a factor of the polynomial.

Once a root is found using the Rational Root Theorem, the Factor Theorem can be used to divide the polynomial by \(x-a\), which simplifies the equation. This process often leads to a lower degree polynomial, making subsequent calculations to find other roots simpler. Using the Factor Theorem iteratively with polynomial division or synthetic division will break the original polynomial into all of its linear factors, revealing all real solutions.